In this section we evaluate the thresholds of the tailored surface code, XZZX code, and surface code in terms of the biased dephasing error model, including the depolarization error model. In particular, when the bias is in \(\eta = \lbrace  0.5, 1, 10, 100, 1000,\infty \rbrace  \) , we obtain the thresholds of the tailored surface code, XZZX code, and surface code on the lattice and heavy-hexagon structure. To evaluate the thresholds, we use the measurement circuit for the syndrome described in the previous section and the stim code[1]} package. In addition, to obtain the correction operator, we use PyMatching[2]}, considering the X and Y errors independently in the tailored surface code and the X and Z errors independently in the XZZX code and surface code. In the decoder, we do not consider the correlation between the errors. In the cases of the surface code and XZZX code, a logical X error and a logical Z error exist. In the tailored surface code, the logical X and Y errors exist. We obtain the logical failure by evaluating the similarity between the logical state to initialized state of the data qubit and the final state applied by the correction operator. For instance, in the surface code, the logical X error rate is evaluated by checking whether the corrected state is the same as \(|{0}\rangle _L\)  when the initialized state is prepared as \(|{0}\rangle _L\) . Meanwhile, the logical Z error rate is evaluated by checking whether the corrected state is the same as \(|{+}\rangle _L\)  when the initialized state is prepared as \(|{+}\rangle _L\) . In QEC code, to consider every error of two logical operators, \(E_{total}\)  is obtained by the logical error rates \(E_{X}\)  and \(E_{Z}\)  as follows[3]}:
\(\begin{array}{cc}E_{total} = 1-(1- E_{X} ) \times (1-E_{Z} ) \\\end{array}\) 
